I'm shooting at ISO 800 aswell not sure why the recommended ISO is 2000 on a side note... I'm sure there's a good reason but I'm from the old school train of thought of use the lowest ISO to get the lowest amount of noise...

For some reason that didn't give you a full screen option when I uploaded that link and without going full screen you can't really see the grain. I've put a link below with some brackets and information in the brackets just delete that information and paste the link into your URL bar and check out the footage there.

So first of all, the native ISO for the FS7 is 2000; that's the lowest noise you'll get. Shooting at 800 is just a way to "rate" the virtual film; 2000 is your sweet spot, and it's where you'll get the full 14 stop dynamic range.

Looking at the footage, it doesn't appear that your LUT is actually applied. What does it look like after grading? These noisy darks may go away once a 709 gamma curve is applied.

Are you on firmware version 3.1? Prior to that firmware version, the camera's noise reduction was only available in Custom mode, not CineEI. With noise reduction off, CineEI mode tends to be noisier in the shadows. One way around this is to overexpose by about a stop; the resulting extra shadow detail allows you to crush the noise in post immediately. Of course, even better than this is upgrading to Firmware 3.1 and just turning on NR.

I highly recommend joining the Sony FS7 user community--there's not a ton of chit chat, but they are pretty quick to respond with help when asked, and there are buckets of threads addressing similar issues to yours.

A couple of reasons; first, this shot is just plain underexposed by about a stop. In S-Log3, your whites should be around 72-78 IRE, and on my scope, they are sitting around 60 before a LUT is applied. When I apply the Sony S-Log3 - > LC709 LUT, your absolute brightest point (the reflections from the lights on the water bottle on the left) are sitting at no higher than 80 IRE; the white board is at 70 IRE. The space under the desk is at around 3-5 IRE. To get it properly exposed, I lifted it by about a stop, which still leaves your white board in the back at around 83 IRE.

Since the area under the desk is underexposed in the first place, you are going to see more noise there when the contrast is reduced--and that's what a Log gamma is; a reduced contrast capture format which will appear like lifted shadows in this case.

So yes, properly exposed S-Log footage will not be noisy like this. But this is underexposed, shot in a lower EI than optimal (especially for low-light environments), and without NR. If all of these things were lined up, this footage would be as clean as Arri footage.

If you're not shooting with a histogram on-screen, this is a SUPER easy trap to fall into when shooting S-Log3. I have made this mistake several times--the image looks washed-out, so you drop exposure to increase contrast...but you end up underexposing as a result. The histogram is absolutely crucial--and if you haven't already, look up some of the guides to exposing for S-Log3; it's very different from S-Log2 or Arri-log.

When shooting in an environment like this, I would tend to overexpose by about a stop; that will still give me plenty of headroom on the whites, but it will give me extra shadow detail that will help me crush any noise in the grade.

And remember, you were shooting in a lower EI than the native; this slightly reduces the camera's dynamic range in the first place. I don't really know from experience how much it affects the noise of the image to shoot an EI this low, but because it resulted in underexposure, plus a slightly reduced dynamic range, with the added bonus of no NR in CineEI mode...you have a few factors in play, all of which equal more noise in the shadows.

TL;DR: when in a darker environment, overexpose a bit to preserve shadow detail. Shoot at the native EI/ISO of 2000. Get your firmware updated so you can use NR in CineEI mode. All of these things will result in reduced shadow noise!